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Theorem logsqvma2 22772
Description: The Möbius inverse of logsqvma 22771. Equation 10.4.8 of [Shapiro], p. 418. (Contributed by Mario Carneiro, 13-May-2016.)
Assertion
Ref Expression
logsqvma2  |-  ( N  e.  NN  ->  sum_ d  e.  { x  e.  NN  |  x  ||  N } 
( ( mmu `  d )  x.  (
( log `  ( N  /  d ) ) ^ 2 ) )  =  ( sum_ d  e.  { x  e.  NN  |  x  ||  N } 
( (Λ `  d )  x.  (Λ `  ( N  /  d ) ) )  +  ( (Λ `  N )  x.  ( log `  N ) ) ) )
Distinct variable group:    x, d, N

Proof of Theorem logsqvma2
Dummy variables  i 
j  k  m  n are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fzfid 11787 . . . . . . . . . 10  |-  ( k  e.  NN  ->  (
1 ... k )  e. 
Fin )
2 sgmss 22424 . . . . . . . . . 10  |-  ( k  e.  NN  ->  { x  e.  NN  |  x  ||  k }  C_  ( 1 ... k ) )
3 ssfi 7525 . . . . . . . . . 10  |-  ( ( ( 1 ... k
)  e.  Fin  /\  { x  e.  NN  |  x  ||  k }  C_  ( 1 ... k
) )  ->  { x  e.  NN  |  x  ||  k }  e.  Fin )
41, 2, 3syl2anc 661 . . . . . . . . 9  |-  ( k  e.  NN  ->  { x  e.  NN  |  x  ||  k }  e.  Fin )
5 ssrab2 3432 . . . . . . . . . . . 12  |-  { x  e.  NN  |  x  ||  k }  C_  NN
6 simpr 461 . . . . . . . . . . . 12  |-  ( ( k  e.  NN  /\  d  e.  { x  e.  NN  |  x  ||  k } )  ->  d  e.  { x  e.  NN  |  x  ||  k } )
75, 6sseldi 3349 . . . . . . . . . . 11  |-  ( ( k  e.  NN  /\  d  e.  { x  e.  NN  |  x  ||  k } )  ->  d  e.  NN )
8 vmacl 22436 . . . . . . . . . . 11  |-  ( d  e.  NN  ->  (Λ `  d )  e.  RR )
97, 8syl 16 . . . . . . . . . 10  |-  ( ( k  e.  NN  /\  d  e.  { x  e.  NN  |  x  ||  k } )  ->  (Λ `  d )  e.  RR )
10 dvdsdivcl 22501 . . . . . . . . . . . 12  |-  ( ( k  e.  NN  /\  d  e.  { x  e.  NN  |  x  ||  k } )  ->  (
k  /  d )  e.  { x  e.  NN  |  x  ||  k } )
115, 10sseldi 3349 . . . . . . . . . . 11  |-  ( ( k  e.  NN  /\  d  e.  { x  e.  NN  |  x  ||  k } )  ->  (
k  /  d )  e.  NN )
12 vmacl 22436 . . . . . . . . . . 11  |-  ( ( k  /  d )  e.  NN  ->  (Λ `  ( k  /  d
) )  e.  RR )
1311, 12syl 16 . . . . . . . . . 10  |-  ( ( k  e.  NN  /\  d  e.  { x  e.  NN  |  x  ||  k } )  ->  (Λ `  ( k  /  d
) )  e.  RR )
149, 13remulcld 9406 . . . . . . . . 9  |-  ( ( k  e.  NN  /\  d  e.  { x  e.  NN  |  x  ||  k } )  ->  (
(Λ `  d )  x.  (Λ `  ( k  /  d ) ) )  e.  RR )
154, 14fsumrecl 13203 . . . . . . . 8  |-  ( k  e.  NN  ->  sum_ d  e.  { x  e.  NN  |  x  ||  k }  ( (Λ `  d
)  x.  (Λ `  (
k  /  d ) ) )  e.  RR )
16 vmacl 22436 . . . . . . . . 9  |-  ( k  e.  NN  ->  (Λ `  k )  e.  RR )
17 nnrp 10992 . . . . . . . . . 10  |-  ( k  e.  NN  ->  k  e.  RR+ )
1817relogcld 22052 . . . . . . . . 9  |-  ( k  e.  NN  ->  ( log `  k )  e.  RR )
1916, 18remulcld 9406 . . . . . . . 8  |-  ( k  e.  NN  ->  (
(Λ `  k )  x.  ( log `  k
) )  e.  RR )
2015, 19readdcld 9405 . . . . . . 7  |-  ( k  e.  NN  ->  ( sum_ d  e.  { x  e.  NN  |  x  ||  k }  ( (Λ `  d )  x.  (Λ `  ( k  /  d
) ) )  +  ( (Λ `  k
)  x.  ( log `  k ) ) )  e.  RR )
2120recnd 9404 . . . . . 6  |-  ( k  e.  NN  ->  ( sum_ d  e.  { x  e.  NN  |  x  ||  k }  ( (Λ `  d )  x.  (Λ `  ( k  /  d
) ) )  +  ( (Λ `  k
)  x.  ( log `  k ) ) )  e.  CC )
2221adantl 466 . . . . 5  |-  ( ( N  e.  NN  /\  k  e.  NN )  ->  ( sum_ d  e.  {
x  e.  NN  |  x  ||  k }  (
(Λ `  d )  x.  (Λ `  ( k  /  d ) ) )  +  ( (Λ `  k )  x.  ( log `  k ) ) )  e.  CC )
23 eqid 2438 . . . . 5  |-  ( k  e.  NN  |->  ( sum_ d  e.  { x  e.  NN  |  x  ||  k }  ( (Λ `  d )  x.  (Λ `  ( k  /  d
) ) )  +  ( (Λ `  k
)  x.  ( log `  k ) ) ) )  =  ( k  e.  NN  |->  ( sum_ d  e.  { x  e.  NN  |  x  ||  k }  ( (Λ `  d )  x.  (Λ `  ( k  /  d
) ) )  +  ( (Λ `  k
)  x.  ( log `  k ) ) ) )
2422, 23fmptd 5862 . . . 4  |-  ( N  e.  NN  ->  (
k  e.  NN  |->  (
sum_ d  e.  {
x  e.  NN  |  x  ||  k }  (
(Λ `  d )  x.  (Λ `  ( k  /  d ) ) )  +  ( (Λ `  k )  x.  ( log `  k ) ) ) ) : NN --> CC )
25 ssrab2 3432 . . . . . . . . 9  |-  { x  e.  NN  |  x  ||  n }  C_  NN
26 simpr 461 . . . . . . . . 9  |-  ( ( ( N  e.  NN  /\  n  e.  NN )  /\  m  e.  {
x  e.  NN  |  x  ||  n } )  ->  m  e.  {
x  e.  NN  |  x  ||  n } )
2725, 26sseldi 3349 . . . . . . . 8  |-  ( ( ( N  e.  NN  /\  n  e.  NN )  /\  m  e.  {
x  e.  NN  |  x  ||  n } )  ->  m  e.  NN )
28 breq2 4291 . . . . . . . . . . . 12  |-  ( k  =  m  ->  (
x  ||  k  <->  x  ||  m
) )
2928rabbidv 2959 . . . . . . . . . . 11  |-  ( k  =  m  ->  { x  e.  NN  |  x  ||  k }  =  {
x  e.  NN  |  x  ||  m } )
30 oveq1 6093 . . . . . . . . . . . . . 14  |-  ( k  =  m  ->  (
k  /  d )  =  ( m  / 
d ) )
3130fveq2d 5690 . . . . . . . . . . . . 13  |-  ( k  =  m  ->  (Λ `  ( k  /  d
) )  =  (Λ `  ( m  /  d
) ) )
3231oveq2d 6102 . . . . . . . . . . . 12  |-  ( k  =  m  ->  (
(Λ `  d )  x.  (Λ `  ( k  /  d ) ) )  =  ( (Λ `  d )  x.  (Λ `  ( m  /  d
) ) ) )
3332adantr 465 . . . . . . . . . . 11  |-  ( ( k  =  m  /\  d  e.  { x  e.  NN  |  x  ||  k } )  ->  (
(Λ `  d )  x.  (Λ `  ( k  /  d ) ) )  =  ( (Λ `  d )  x.  (Λ `  ( m  /  d
) ) ) )
3429, 33sumeq12dv 13175 . . . . . . . . . 10  |-  ( k  =  m  ->  sum_ d  e.  { x  e.  NN  |  x  ||  k }  ( (Λ `  d
)  x.  (Λ `  (
k  /  d ) ) )  =  sum_ d  e.  { x  e.  NN  |  x  ||  m }  ( (Λ `  d )  x.  (Λ `  ( m  /  d
) ) ) )
35 fveq2 5686 . . . . . . . . . . 11  |-  ( k  =  m  ->  (Λ `  k )  =  (Λ `  m ) )
36 fveq2 5686 . . . . . . . . . . 11  |-  ( k  =  m  ->  ( log `  k )  =  ( log `  m
) )
3735, 36oveq12d 6104 . . . . . . . . . 10  |-  ( k  =  m  ->  (
(Λ `  k )  x.  ( log `  k
) )  =  ( (Λ `  m )  x.  ( log `  m
) ) )
3834, 37oveq12d 6104 . . . . . . . . 9  |-  ( k  =  m  ->  ( sum_ d  e.  { x  e.  NN  |  x  ||  k }  ( (Λ `  d )  x.  (Λ `  ( k  /  d
) ) )  +  ( (Λ `  k
)  x.  ( log `  k ) ) )  =  ( sum_ d  e.  { x  e.  NN  |  x  ||  m } 
( (Λ `  d )  x.  (Λ `  ( m  /  d ) ) )  +  ( (Λ `  m )  x.  ( log `  m ) ) ) )
39 ovex 6111 . . . . . . . . 9  |-  ( sum_ d  e.  { x  e.  NN  |  x  ||  k }  ( (Λ `  d )  x.  (Λ `  ( k  /  d
) ) )  +  ( (Λ `  k
)  x.  ( log `  k ) ) )  e.  _V
4038, 23, 39fvmpt3i 5773 . . . . . . . 8  |-  ( m  e.  NN  ->  (
( k  e.  NN  |->  ( sum_ d  e.  {
x  e.  NN  |  x  ||  k }  (
(Λ `  d )  x.  (Λ `  ( k  /  d ) ) )  +  ( (Λ `  k )  x.  ( log `  k ) ) ) ) `  m
)  =  ( sum_ d  e.  { x  e.  NN  |  x  ||  m }  ( (Λ `  d )  x.  (Λ `  ( m  /  d
) ) )  +  ( (Λ `  m
)  x.  ( log `  m ) ) ) )
4127, 40syl 16 . . . . . . 7  |-  ( ( ( N  e.  NN  /\  n  e.  NN )  /\  m  e.  {
x  e.  NN  |  x  ||  n } )  ->  ( ( k  e.  NN  |->  ( sum_ d  e.  { x  e.  NN  |  x  ||  k }  ( (Λ `  d )  x.  (Λ `  ( k  /  d
) ) )  +  ( (Λ `  k
)  x.  ( log `  k ) ) ) ) `  m )  =  ( sum_ d  e.  { x  e.  NN  |  x  ||  m } 
( (Λ `  d )  x.  (Λ `  ( m  /  d ) ) )  +  ( (Λ `  m )  x.  ( log `  m ) ) ) )
4241sumeq2dv 13172 . . . . . 6  |-  ( ( N  e.  NN  /\  n  e.  NN )  -> 
sum_ m  e.  { x  e.  NN  |  x  ||  n }  ( (
k  e.  NN  |->  (
sum_ d  e.  {
x  e.  NN  |  x  ||  k }  (
(Λ `  d )  x.  (Λ `  ( k  /  d ) ) )  +  ( (Λ `  k )  x.  ( log `  k ) ) ) ) `  m
)  =  sum_ m  e.  { x  e.  NN  |  x  ||  n } 
( sum_ d  e.  {
x  e.  NN  |  x  ||  m }  (
(Λ `  d )  x.  (Λ `  ( m  /  d ) ) )  +  ( (Λ `  m )  x.  ( log `  m ) ) ) )
43 logsqvma 22771 . . . . . . 7  |-  ( n  e.  NN  ->  sum_ m  e.  { x  e.  NN  |  x  ||  n } 
( sum_ d  e.  {
x  e.  NN  |  x  ||  m }  (
(Λ `  d )  x.  (Λ `  ( m  /  d ) ) )  +  ( (Λ `  m )  x.  ( log `  m ) ) )  =  ( ( log `  n ) ^ 2 ) )
4443adantl 466 . . . . . 6  |-  ( ( N  e.  NN  /\  n  e.  NN )  -> 
sum_ m  e.  { x  e.  NN  |  x  ||  n }  ( sum_ d  e.  { x  e.  NN  |  x  ||  m }  ( (Λ `  d )  x.  (Λ `  ( m  /  d
) ) )  +  ( (Λ `  m
)  x.  ( log `  m ) ) )  =  ( ( log `  n ) ^ 2 ) )
4542, 44eqtr2d 2471 . . . . 5  |-  ( ( N  e.  NN  /\  n  e.  NN )  ->  ( ( log `  n
) ^ 2 )  =  sum_ m  e.  {
x  e.  NN  |  x  ||  n }  (
( k  e.  NN  |->  ( sum_ d  e.  {
x  e.  NN  |  x  ||  k }  (
(Λ `  d )  x.  (Λ `  ( k  /  d ) ) )  +  ( (Λ `  k )  x.  ( log `  k ) ) ) ) `  m
) )
4645mpteq2dva 4373 . . . 4  |-  ( N  e.  NN  ->  (
n  e.  NN  |->  ( ( log `  n
) ^ 2 ) )  =  ( n  e.  NN  |->  sum_ m  e.  { x  e.  NN  |  x  ||  n } 
( ( k  e.  NN  |->  ( sum_ d  e.  { x  e.  NN  |  x  ||  k }  ( (Λ `  d
)  x.  (Λ `  (
k  /  d ) ) )  +  ( (Λ `  k )  x.  ( log `  k
) ) ) ) `
 m ) ) )
4724, 46muinv 22513 . . 3  |-  ( N  e.  NN  ->  (
k  e.  NN  |->  (
sum_ d  e.  {
x  e.  NN  |  x  ||  k }  (
(Λ `  d )  x.  (Λ `  ( k  /  d ) ) )  +  ( (Λ `  k )  x.  ( log `  k ) ) ) )  =  ( i  e.  NN  |->  sum_ j  e.  { x  e.  NN  |  x  ||  i }  ( (
mmu `  j )  x.  ( ( n  e.  NN  |->  ( ( log `  n ) ^ 2 ) ) `  (
i  /  j ) ) ) ) )
4847fveq1d 5688 . 2  |-  ( N  e.  NN  ->  (
( k  e.  NN  |->  ( sum_ d  e.  {
x  e.  NN  |  x  ||  k }  (
(Λ `  d )  x.  (Λ `  ( k  /  d ) ) )  +  ( (Λ `  k )  x.  ( log `  k ) ) ) ) `  N
)  =  ( ( i  e.  NN  |->  sum_ j  e.  { x  e.  NN  |  x  ||  i }  ( (
mmu `  j )  x.  ( ( n  e.  NN  |->  ( ( log `  n ) ^ 2 ) ) `  (
i  /  j ) ) ) ) `  N ) )
49 breq2 4291 . . . . . 6  |-  ( k  =  N  ->  (
x  ||  k  <->  x  ||  N
) )
5049rabbidv 2959 . . . . 5  |-  ( k  =  N  ->  { x  e.  NN  |  x  ||  k }  =  {
x  e.  NN  |  x  ||  N } )
51 oveq1 6093 . . . . . . . 8  |-  ( k  =  N  ->  (
k  /  d )  =  ( N  / 
d ) )
5251fveq2d 5690 . . . . . . 7  |-  ( k  =  N  ->  (Λ `  ( k  /  d
) )  =  (Λ `  ( N  /  d
) ) )
5352oveq2d 6102 . . . . . 6  |-  ( k  =  N  ->  (
(Λ `  d )  x.  (Λ `  ( k  /  d ) ) )  =  ( (Λ `  d )  x.  (Λ `  ( N  /  d
) ) ) )
5453adantr 465 . . . . 5  |-  ( ( k  =  N  /\  d  e.  { x  e.  NN  |  x  ||  k } )  ->  (
(Λ `  d )  x.  (Λ `  ( k  /  d ) ) )  =  ( (Λ `  d )  x.  (Λ `  ( N  /  d
) ) ) )
5550, 54sumeq12dv 13175 . . . 4  |-  ( k  =  N  ->  sum_ d  e.  { x  e.  NN  |  x  ||  k }  ( (Λ `  d
)  x.  (Λ `  (
k  /  d ) ) )  =  sum_ d  e.  { x  e.  NN  |  x  ||  N }  ( (Λ `  d )  x.  (Λ `  ( N  /  d
) ) ) )
56 fveq2 5686 . . . . 5  |-  ( k  =  N  ->  (Λ `  k )  =  (Λ `  N ) )
57 fveq2 5686 . . . . 5  |-  ( k  =  N  ->  ( log `  k )  =  ( log `  N
) )
5856, 57oveq12d 6104 . . . 4  |-  ( k  =  N  ->  (
(Λ `  k )  x.  ( log `  k
) )  =  ( (Λ `  N )  x.  ( log `  N
) ) )
5955, 58oveq12d 6104 . . 3  |-  ( k  =  N  ->  ( sum_ d  e.  { x  e.  NN  |  x  ||  k }  ( (Λ `  d )  x.  (Λ `  ( k  /  d
) ) )  +  ( (Λ `  k
)  x.  ( log `  k ) ) )  =  ( sum_ d  e.  { x  e.  NN  |  x  ||  N } 
( (Λ `  d )  x.  (Λ `  ( N  /  d ) ) )  +  ( (Λ `  N )  x.  ( log `  N ) ) ) )
6059, 23, 39fvmpt3i 5773 . 2  |-  ( N  e.  NN  ->  (
( k  e.  NN  |->  ( sum_ d  e.  {
x  e.  NN  |  x  ||  k }  (
(Λ `  d )  x.  (Λ `  ( k  /  d ) ) )  +  ( (Λ `  k )  x.  ( log `  k ) ) ) ) `  N
)  =  ( sum_ d  e.  { x  e.  NN  |  x  ||  N }  ( (Λ `  d )  x.  (Λ `  ( N  /  d
) ) )  +  ( (Λ `  N
)  x.  ( log `  N ) ) ) )
61 fveq2 5686 . . . . . 6  |-  ( j  =  d  ->  (
mmu `  j )  =  ( mmu `  d ) )
62 oveq2 6094 . . . . . . . 8  |-  ( j  =  d  ->  (
i  /  j )  =  ( i  / 
d ) )
6362fveq2d 5690 . . . . . . 7  |-  ( j  =  d  ->  ( log `  ( i  / 
j ) )  =  ( log `  (
i  /  d ) ) )
6463oveq1d 6101 . . . . . 6  |-  ( j  =  d  ->  (
( log `  (
i  /  j ) ) ^ 2 )  =  ( ( log `  ( i  /  d
) ) ^ 2 ) )
6561, 64oveq12d 6104 . . . . 5  |-  ( j  =  d  ->  (
( mmu `  j
)  x.  ( ( log `  ( i  /  j ) ) ^ 2 ) )  =  ( ( mmu `  d )  x.  (
( log `  (
i  /  d ) ) ^ 2 ) ) )
6665cbvsumv 13165 . . . 4  |-  sum_ j  e.  { x  e.  NN  |  x  ||  i }  ( ( mmu `  j )  x.  (
( log `  (
i  /  j ) ) ^ 2 ) )  =  sum_ d  e.  { x  e.  NN  |  x  ||  i }  ( ( mmu `  d )  x.  (
( log `  (
i  /  d ) ) ^ 2 ) )
67 breq2 4291 . . . . . 6  |-  ( i  =  N  ->  (
x  ||  i  <->  x  ||  N
) )
6867rabbidv 2959 . . . . 5  |-  ( i  =  N  ->  { x  e.  NN  |  x  ||  i }  =  {
x  e.  NN  |  x  ||  N } )
69 oveq1 6093 . . . . . . . . 9  |-  ( i  =  N  ->  (
i  /  d )  =  ( N  / 
d ) )
7069fveq2d 5690 . . . . . . . 8  |-  ( i  =  N  ->  ( log `  ( i  / 
d ) )  =  ( log `  ( N  /  d ) ) )
7170oveq1d 6101 . . . . . . 7  |-  ( i  =  N  ->  (
( log `  (
i  /  d ) ) ^ 2 )  =  ( ( log `  ( N  /  d
) ) ^ 2 ) )
7271oveq2d 6102 . . . . . 6  |-  ( i  =  N  ->  (
( mmu `  d
)  x.  ( ( log `  ( i  /  d ) ) ^ 2 ) )  =  ( ( mmu `  d )  x.  (
( log `  ( N  /  d ) ) ^ 2 ) ) )
7372adantr 465 . . . . 5  |-  ( ( i  =  N  /\  d  e.  { x  e.  NN  |  x  ||  i } )  ->  (
( mmu `  d
)  x.  ( ( log `  ( i  /  d ) ) ^ 2 ) )  =  ( ( mmu `  d )  x.  (
( log `  ( N  /  d ) ) ^ 2 ) ) )
7468, 73sumeq12dv 13175 . . . 4  |-  ( i  =  N  ->  sum_ d  e.  { x  e.  NN  |  x  ||  i }  ( ( mmu `  d )  x.  (
( log `  (
i  /  d ) ) ^ 2 ) )  =  sum_ d  e.  { x  e.  NN  |  x  ||  N } 
( ( mmu `  d )  x.  (
( log `  ( N  /  d ) ) ^ 2 ) ) )
7566, 74syl5eq 2482 . . 3  |-  ( i  =  N  ->  sum_ j  e.  { x  e.  NN  |  x  ||  i }  ( ( mmu `  j )  x.  (
( log `  (
i  /  j ) ) ^ 2 ) )  =  sum_ d  e.  { x  e.  NN  |  x  ||  N } 
( ( mmu `  d )  x.  (
( log `  ( N  /  d ) ) ^ 2 ) ) )
76 ssrab2 3432 . . . . . . . 8  |-  { x  e.  NN  |  x  ||  i }  C_  NN
77 dvdsdivcl 22501 . . . . . . . 8  |-  ( ( i  e.  NN  /\  j  e.  { x  e.  NN  |  x  ||  i } )  ->  (
i  /  j )  e.  { x  e.  NN  |  x  ||  i } )
7876, 77sseldi 3349 . . . . . . 7  |-  ( ( i  e.  NN  /\  j  e.  { x  e.  NN  |  x  ||  i } )  ->  (
i  /  j )  e.  NN )
79 fveq2 5686 . . . . . . . . 9  |-  ( n  =  ( i  / 
j )  ->  ( log `  n )  =  ( log `  (
i  /  j ) ) )
8079oveq1d 6101 . . . . . . . 8  |-  ( n  =  ( i  / 
j )  ->  (
( log `  n
) ^ 2 )  =  ( ( log `  ( i  /  j
) ) ^ 2 ) )
81 eqid 2438 . . . . . . . 8  |-  ( n  e.  NN  |->  ( ( log `  n ) ^ 2 ) )  =  ( n  e.  NN  |->  ( ( log `  n ) ^ 2 ) )
82 ovex 6111 . . . . . . . 8  |-  ( ( log `  n ) ^ 2 )  e. 
_V
8380, 81, 82fvmpt3i 5773 . . . . . . 7  |-  ( ( i  /  j )  e.  NN  ->  (
( n  e.  NN  |->  ( ( log `  n
) ^ 2 ) ) `  ( i  /  j ) )  =  ( ( log `  ( i  /  j
) ) ^ 2 ) )
8478, 83syl 16 . . . . . 6  |-  ( ( i  e.  NN  /\  j  e.  { x  e.  NN  |  x  ||  i } )  ->  (
( n  e.  NN  |->  ( ( log `  n
) ^ 2 ) ) `  ( i  /  j ) )  =  ( ( log `  ( i  /  j
) ) ^ 2 ) )
8584oveq2d 6102 . . . . 5  |-  ( ( i  e.  NN  /\  j  e.  { x  e.  NN  |  x  ||  i } )  ->  (
( mmu `  j
)  x.  ( ( n  e.  NN  |->  ( ( log `  n
) ^ 2 ) ) `  ( i  /  j ) ) )  =  ( ( mmu `  j )  x.  ( ( log `  ( i  /  j
) ) ^ 2 ) ) )
8685sumeq2dv 13172 . . . 4  |-  ( i  e.  NN  ->  sum_ j  e.  { x  e.  NN  |  x  ||  i }  ( ( mmu `  j )  x.  (
( n  e.  NN  |->  ( ( log `  n
) ^ 2 ) ) `  ( i  /  j ) ) )  =  sum_ j  e.  { x  e.  NN  |  x  ||  i }  ( ( mmu `  j )  x.  (
( log `  (
i  /  j ) ) ^ 2 ) ) )
8786mpteq2ia 4369 . . 3  |-  ( i  e.  NN  |->  sum_ j  e.  { x  e.  NN  |  x  ||  i }  ( ( mmu `  j )  x.  (
( n  e.  NN  |->  ( ( log `  n
) ^ 2 ) ) `  ( i  /  j ) ) ) )  =  ( i  e.  NN  |->  sum_ j  e.  { x  e.  NN  |  x  ||  i }  ( (
mmu `  j )  x.  ( ( log `  (
i  /  j ) ) ^ 2 ) ) )
88 sumex 13157 . . 3  |-  sum_ j  e.  { x  e.  NN  |  x  ||  i }  ( ( mmu `  j )  x.  (
( log `  (
i  /  j ) ) ^ 2 ) )  e.  _V
8975, 87, 88fvmpt3i 5773 . 2  |-  ( N  e.  NN  ->  (
( i  e.  NN  |->  sum_ j  e.  { x  e.  NN  |  x  ||  i }  ( (
mmu `  j )  x.  ( ( n  e.  NN  |->  ( ( log `  n ) ^ 2 ) ) `  (
i  /  j ) ) ) ) `  N )  =  sum_ d  e.  { x  e.  NN  |  x  ||  N }  ( (
mmu `  d )  x.  ( ( log `  ( N  /  d ) ) ^ 2 ) ) )
9048, 60, 893eqtr3rd 2479 1  |-  ( N  e.  NN  ->  sum_ d  e.  { x  e.  NN  |  x  ||  N } 
( ( mmu `  d )  x.  (
( log `  ( N  /  d ) ) ^ 2 ) )  =  ( sum_ d  e.  { x  e.  NN  |  x  ||  N } 
( (Λ `  d )  x.  (Λ `  ( N  /  d ) ) )  +  ( (Λ `  N )  x.  ( log `  N ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756   {crab 2714    C_ wss 3323   class class class wbr 4287    e. cmpt 4345   ` cfv 5413  (class class class)co 6086   Fincfn 7302   CCcc 9272   RRcr 9273   1c1 9275    + caddc 9277    x. cmul 9279    / cdiv 9985   NNcn 10314   2c2 10363   ...cfz 11429   ^cexp 11857   sum_csu 13155    || cdivides 13527   logclog 21986  Λcvma 22409   mmucmu 22412
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-rep 4398  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367  ax-inf2 7839  ax-cnex 9330  ax-resscn 9331  ax-1cn 9332  ax-icn 9333  ax-addcl 9334  ax-addrcl 9335  ax-mulcl 9336  ax-mulrcl 9337  ax-mulcom 9338  ax-addass 9339  ax-mulass 9340  ax-distr 9341  ax-i2m1 9342  ax-1ne0 9343  ax-1rid 9344  ax-rnegex 9345  ax-rrecex 9346  ax-cnre 9347  ax-pre-lttri 9348  ax-pre-lttrn 9349  ax-pre-ltadd 9350  ax-pre-mulgt0 9351  ax-pre-sup 9352  ax-addf 9353  ax-mulf 9354
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2715  df-rex 2716  df-reu 2717  df-rmo 2718  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-pss 3339  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-tp 3877  df-op 3879  df-uni 4087  df-int 4124  df-iun 4168  df-iin 4169  df-disj 4258  df-br 4288  df-opab 4346  df-mpt 4347  df-tr 4381  df-eprel 4627  df-id 4631  df-po 4636  df-so 4637  df-fr 4674  df-se 4675  df-we 4676  df-ord 4717  df-on 4718  df-lim 4719  df-suc 4720  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-riota 6047  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-of 6315  df-om 6472  df-1st 6572  df-2nd 6573  df-supp 6686  df-recs 6824  df-rdg 6858  df-1o 6912  df-2o 6913  df-oadd 6916  df-er 7093  df-map 7208  df-pm 7209  df-ixp 7256  df-en 7303  df-dom 7304  df-sdom 7305  df-fin 7306  df-fsupp 7613  df-fi 7653  df-sup 7683  df-oi 7716  df-card 8101  df-cda 8329  df-pnf 9412  df-mnf 9413  df-xr 9414  df-ltxr 9415  df-le 9416  df-sub 9589  df-neg 9590  df-div 9986  df-nn 10315  df-2 10372  df-3 10373  df-4 10374  df-5 10375  df-6 10376  df-7 10377  df-8 10378  df-9 10379  df-10 10380  df-n0 10572  df-z 10639  df-dec 10748  df-uz 10854  df-q 10946  df-rp 10984  df-xneg 11081  df-xadd 11082  df-xmul 11083  df-ioo 11296  df-ioc 11297  df-ico 11298  df-icc 11299  df-fz 11430  df-fzo 11541  df-fl 11634  df-mod 11701  df-seq 11799  df-exp 11858  df-fac 12044  df-bc 12071  df-hash 12096  df-shft 12548  df-cj 12580  df-re 12581  df-im 12582  df-sqr 12716  df-abs 12717  df-limsup 12941  df-clim 12958  df-rlim 12959  df-sum 13156  df-ef 13345  df-sin 13347  df-cos 13348  df-pi 13350  df-dvds 13528  df-gcd 13683  df-prm 13756  df-pc 13896  df-struct 14168  df-ndx 14169  df-slot 14170  df-base 14171  df-sets 14172  df-ress 14173  df-plusg 14243  df-mulr 14244  df-starv 14245  df-sca 14246  df-vsca 14247  df-ip 14248  df-tset 14249  df-ple 14250  df-ds 14252  df-unif 14253  df-hom 14254  df-cco 14255  df-rest 14353  df-topn 14354  df-0g 14372  df-gsum 14373  df-topgen 14374  df-pt 14375  df-prds 14378  df-xrs 14432  df-qtop 14437  df-imas 14438  df-xps 14440  df-mre 14516  df-mrc 14517  df-acs 14519  df-mnd 15407  df-submnd 15457  df-mulg 15539  df-cntz 15826  df-cmn 16270  df-psmet 17789  df-xmet 17790  df-met 17791  df-bl 17792  df-mopn 17793  df-fbas 17794  df-fg 17795  df-cnfld 17799  df-top 18483  df-bases 18485  df-topon 18486  df-topsp 18487  df-cld 18603  df-ntr 18604  df-cls 18605  df-nei 18682  df-lp 18720  df-perf 18721  df-cn 18811  df-cnp 18812  df-haus 18899  df-tx 19115  df-hmeo 19308  df-fil 19399  df-fm 19491  df-flim 19492  df-flf 19493  df-xms 19875  df-ms 19876  df-tms 19877  df-cncf 20434  df-limc 21321  df-dv 21322  df-log 21988  df-vma 22415  df-mu 22418
This theorem is referenced by:  selberg  22777
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